282 X EXISTENCE THEOREMS

that g(B) <= U and f(g(z)) = z for z e B. (Consider first the case in which / is analytic
in a neighborhood of 0, and take for z0 a point where |/'(z)|(l - |z|2) reaches its
maximum; use then (d) to reduce the problem to the case in which z0 = 0, and apply
in that case the result of (c) to a function of the form a +/(Rz), where a and R are
suitable complex numbers. In the general case consider the function /((I — e)z)/(l — e),
where e > 0 is arbitrarily small.)

6. (a) Let W be the set of all complex valued functions / analytic in the unit disk U:
z| < 1, such that/(U) does not contain the points 0 and 1. For any f unction /e9ft,
there is a unique analytic function g in U such that exp(2?n#(z)) =/(z) in U and
|^(0(0))| < IT (Section 10.2, Problem 7); g(\J) does not contain any positive or negative
integer. Furthermore (same reference) there is an analytic function h in U such that
g(z)l(g(z) - 1) = ((1 + h(z))l(\ - h(z)))2; h(V) does not contain any of the points 0, 1,
c'n = (Vn+Vn--~l)2 and c; = (\/«"-V«- I)2 (n integer ^1). Finally, there is an
analytic function 99 in U such that exp(<p(z)) = A(z); <p(U) does not contain any of the
points log c'n + 2&7J7, log cn + 1km (k positive or negative integer, n ^ 1). Show that
no disk of radius >4 can be contained in <p(U); using Problem 5(e), deduce from that
result that

for |jc|<l (consider the function t-*ccp(x + (l - \x\)t)9 for a suitably chosen
constant c). Conclude that there is a function F(u, v), finite and continuous in
(C- {0, 1}) x [0, 1[, such that for every function /e 9R, log|/(z)| ^ F(/(0),r) for
any |z| ^r< 1.

(b) Let fe W be such that either |/(0)| < 1/2 or |/(0) - 1 1 < 1/2. Given r such that
0 ^ r < 1, show that either |/(z)| ^ 5/2 for |z| < r, or there exists a point x such that
|*| < r and |/(*)| ^ 1/2, |/(*)- 1| ^ 1/2 and |1//(*)| ^ 1/2. Applying the result of (a)
to the function f((z - x)/(xz - 1)), conclude that there is a function Fi(z/,u), con-
tinuous and finite in [0, +oo[ x [0, l[, such that for any function /e 9tt, the relations
|/(0)| ^ s and |z| < r imply |/(z)| ^ Fj(y, r) ("Schottky's theorem").

7. Let A be an open connected subset of C, and (/„) a sequence of functions of the set 9ft
(Problem 6). Show that for any compact subset L of A, there exists a subsequence
(/nfc) such that either that subsequence is uniformly convergent in L, or the sequence
(\lfnk) converges uniformly to 0 in L. (Using Schottky's theorem, prove that the points
x e A such that lim (!//„(#)) = 0 form an open and closed subset of A, hence equal to

n-+oo

A or empty; in the second case, show, using the compactness of L, that there is a sub-
sequence of (fn) which is bounded in a compact neighborhood of L, and apply (9.13.1);
in the first case, use similarly (9.13.1) applied to the sequence (!//„).)

8. (a) Let / be a complex valued function, analytic in the open set V : 0 < |z — a < r,
and suppose a is an essential singularity of /(Section 9.1 5). Show that C — /(V) is empty
or reduced to a single point ("Picard's theorem". Let W be the open subset of
V defined by r/2< |z — a\ < r and consider in W the family of analytic functions
fn(z)=f(zl2n); if there are at least two distinct points in C— /(V), apply Problem
7 to the sequence (/„), and derive a contradiction with Problem 2 of Section 9.15, using
(9.15.2).)

(b) Deduce from (a) that if g is an entire function in C, which is not a constant,
then C — #(C) is empty or reduced to a single point (consider g(\lz) in C — • {0}).

9. (a) Show that there is an entire function f(x, y) in C2 satisfying the identity

f(4x, 4y) - 4/(x, y) - -5(/(2x, -2j>))2 + 2(/(2*, ~2y))s

and such that the term of degree ^ 1 in the Taylor development of /at the point (0, 0)
are x + y (Section 10.1, Problem 11).s